Finite Group Databasebeta

← #482144

Subgroups — #482144

Order 512.

977 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35739C211yesmaximal, characteristic
2#20640C211yesmaximal, characteristic
3#17351C211yesmaximal, characteristic
4#11908C212yesmaximal, complemented
5#11906C212yesmaximal, complemented
6#11905C212yesmaximal, complemented
7#11904C212yesmaximal, complemented
8#11722C212yesmaximal, complemented
9C2×#1354C212yesmaximal, complemented
10C2×D4×Q8C411yescharacteristic
11C2×#520C2211yescharacteristic
12#2972—21no—
13#2905—21no—
14C2×#413—21no—
15#2628C411yescharacteristic
16#2627C411yescharacteristic
17#2627C2211yescharacteristic
18#2581C2211yescharacteristic
19#2567C2211yescharacteristic
20#2564C411yescharacteristic
21#2564C2211yescharacteristic
22#2451—21no—
23#2449—21no—
24#2449—21no—
25#2440—21no—
26#2438—21no—
27#2438—21no—
28#2409C411yescharacteristic
29#2409C2211yescharacteristic
30#2396C411yescharacteristic
31C2×#395C411yescharacteristic
32C2×#395C2211yescharacteristic
33#2349C411yescharacteristic
34D4×C4⋊C4—21no—
35#2339—21no—
36#1801—21no—
37#1801C2212yes—
38#1800—21no—
39#1800C2212yes—
40C2×#343C2212yes—
41#1461C2212yes—
42#1459—21no—
43#1459C2212yes—
44#1458—21no—
45C2×#324—21no—
46C2×#324C2212yes—
47#1354C2214yes—
48#1354C2214yes—
49#1354C2214yes—
50#1353C2214yes—
51C22×#41D411yescharacteristic
52C23×Q8C2×C411yescharacteristic
53C23×Q8C2×C411yescharacteristic
54C22×M16—22no—
55D4×Q8—44no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8C2×C411yescharacteristic
63C2×C22⋊Q8C2×C411yescharacteristic
64#520—44no—
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8C2311yescharacteristic
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×#117C2×C411yescharacteristic
77C2×#117D411yescharacteristic
78C2×#117D411yescharacteristic
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C8×D4—44no—
85#431—22no—
86#424—44no—
87#423—22no—
88#423—22no—
89#413—44no—
90C2×#101—22no—
91C2×#100—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94#400—21no—
95#400C2×C411yescharacteristic
96#400C2×C411yescharacteristic
97#400C2311yescharacteristic
98#399—21no—
99#399C2×C411yescharacteristic
100#398—21no—
101#398—21no—
102#398—21no—
103#398—21no—
104#398—21no—
105#396—21no—
106#395—21no—
107#395—21no—
108#395—21no—
109#395—21no—
110#395C2×C411yescharacteristic
111#395C2×C411yescharacteristic
112#395C2×C411yescharacteristic
113#395C2×C411yescharacteristic
114#395C2×C411yescharacteristic
115#395C2×C411yescharacteristic
116#395C2311yescharacteristic
117#395C2311yescharacteristic
118#395C2311yescharacteristic
119#393C2×C411yescharacteristic
120#393C2×C411yescharacteristic
121#393C2×C411yescharacteristic
122#391—21no—
123#391—21no—
124#391—21no—
125#391—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#389—21no—
131#387—21no—
132#387—21no—
133#387—21no—
134#387—21no—
135#386—21no—
136#386—21no—
137#386—21no—
138#386—21no—
139#385—21no—
140#385—21no—
141#385—21no—
142#385—21no—
143#385—21no—
144#385—21no—
145#385—21no—
146#385—21no—
147#384—21no—
148#382C2×C411yescharacteristic
149#381—21no—
150#381—21no—
151#380—21no—
152#380—21no—
153#380—21no—
154#380C2×C411yescharacteristic
155#380C2×C411yescharacteristic
156C4×C4⋊C4—21no—
157C4×C4⋊C4C2×C411yescharacteristic
158C4×C22⋊C4—21no—
159C2×#95—21no—
160C2×#95—21no—
161C2×#95C2×C411yescharacteristic
162C2×#95C2×C411yescharacteristic
163#343—22no—
164#343C2314yes—
165#324—22no—
166#324—22no—
167#324C2314yes—
168#323—22no—
169C2×#41—42no—
170C2×#41—42no—
171C2×#41SD1614yes—
172C2×#41Q1614yes—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—42no—
176C22×Q8—42no—
177C22×Q8—82no—
178C22×Q8—82no—
179C22×Q8C22×C411yescharacteristic
180C22×Q8C22×C411yescharacteristic
181C22×Q8C22×C411yescharacteristic
182C22×Q8C2×D411yescharacteristic
183C22×D4C22⋊C411yescharacteristic
184C22×D4C22⋊C411yescharacteristic
185C22×D4C2×D411yescharacteristic
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22⋊C411yescharacteristic
192C23×C4C22⋊C411yescharacteristic
193C23×C4C22⋊C411yescharacteristic
194C23×C4C22×C411yescharacteristic
195C23×C4C22×C411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C2×M16—24no—
200C2×M16—42no—
201C2×M16—44no—
202C2×M16—44no—
203C22×C8—42no—
204C22×C8—42no—
205C22×C8—42no—
206C4⋊Q8—42no—
207C4⋊Q8—44no—
208C22⋊Q8—44no—
209C22⋊Q8—42no—
210C22⋊Q8—44no—
211C22⋊Q8—42no—
212C22⋊Q8—82no—
213C22⋊Q8—82no—
214C4×Q8—42no—
215C4×Q8—42no—
216C4×Q8—42no—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—42no—
223C4×D4—44no—
224C4×D4—44no—
225#117—24no—
226#117—42no—
227#117—42no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—82no—
251C2×C4⋊C4—82no—
252C2×C4⋊C4C22⋊C411yescharacteristic
253C2×C4⋊C4C22⋊C411yescharacteristic
254C2×C4⋊C4C22⋊C411yescharacteristic
255C2×C4⋊C4C22⋊C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2411yescharacteristic
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4C22⋊C411yescharacteristic
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C22⋊C411yescharacteristic
274C2×C22⋊C4C22⋊C411yescharacteristic
275C2×C22⋊C4C22×C411yescharacteristic
276C2×C22⋊C4C2×D411yescharacteristic
277C2×C42—21no—
278C2×C42—21no—
279C2×C42C22⋊C411yescharacteristic
280C2×C42C22⋊C411yescharacteristic
281C2×C42C22×C411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C4⋊C8—44no—
286C4⋊C8—44no—
287C4⋊C8—42no—
288#101—44no—
289#100—44no—
290C22⋊C8—24no—
291C22⋊C8—42no—
292C22⋊C8—42no—
293C4×C8—42no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95C22×C411yescharacteristic
316#95C22×C411yescharacteristic
317#95C22×C411yescharacteristic
318#95C22×C411yescharacteristic
319C24C2×C22⋊C411yescharacteristic
320C24C2×C22⋊C411yescharacteristic
321C24C2×C22⋊C411yescharacteristic
322#41—48no—
323#41—48no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—42no—
334C2×Q8—81no—
335C2×Q8—81no—
336C2×Q8—81no—
337C2×Q8—81no—
338C2×Q8—84no—
339C2×Q8—84no—
340C2×Q8—84no—
341C2×Q8—84no—
342C2×Q8C2×SD1612yes—
343C2×Q8C2×Q1612yes—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4#10314yes—
351C2×D4#10314yes—
352C2×D4C2×SD1612yes—
353C2×D4C2×Q1612yes—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—81no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4—81no—
392C22×C4#10314yes—
393C22×C4#10314yes—
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×SD1612yes—
404C22×C4C2×Q1612yes—
405C22×C4C23×C411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408M16—44no—
409M16—44no—
410M16—84no—
411C2×C8—44no—
412C2×C8—44no—
413C2×C8—84no—
414C2×C8—162no—
415C2×C8—162no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—81no—
439C4⋊C4—81no—
440C4⋊C4—81no—
441C4⋊C4—81no—
442C4⋊C4—81no—
443C4⋊C4—81no—
444C4⋊C4—81no—
445C4⋊C4—81no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—81no—
458C22⋊C4—81no—
459C22⋊C4#9914yes—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—41no—
465C42—41no—
466C42—42no—
467C42—42no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23C2×#9912yes—
479C23C2×#10312yes—
480C23C2×#10312yes—
481C23C2×#10312yes—
482C23#42112yes—
483C23C22×C22⋊C411yescharacteristic
484Q8—42no—
485Q8—42no—
486Q8—42no—
487Q8—42no—
488Q8—84no—
489Q8—84no—
490Q8—82no—
491Q8—82no—
492D4—28no—
493D4—28no—
494D4—42no—
495D4—42no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—84no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4—161no—
555C2×C4—161no—
556C2×C4C2×#9912yes—
557C2×C4C2×#9912yes—
558C2×C4C2×#10312yes—
559C2×C4C2×#10312yes—
560C2×C4C2×#10312yes—
561C2×C4C22×SD1611yescharacteristic
562C2×C4C22×Q1611yescharacteristic
563C8—84no—
564C8—84no—
565C22—21no—
566C22—22no—
567C22—21no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—81no—
581C22#152014yes—
582C22C22×#9911yescharacteristic
583C22C22×#10311yescharacteristic
584C22C2×#42111yescharacteristic
585C4—22no—
586C4—22no—
587C4—21no—
588C4—22no—
589C4—21no—
590C4—22no—
591C4—22no—
592C4—42no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4—81no—
604C4—81no—
605C4—81no—
606C2—21no—
607C2—21no—
608C2—24no—
609C2—42no—
610C2—42no—
611C2C2×#152011yescharacteristic
612C2#3334811yescharacteristic
613C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #11908C2semidirect
5. #11906C2semidirect
6. #11905C2semidirect
7. #11904C2semidirect
8. #11722C2semidirect
9. C2×#1354C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#357392
#206402
#173512
#119082
#119062
#119052
#119042
#117222
C2×#13542